Find X: Matrix Determinant Of 9

Hey guys! Ever stared at a matrix and wondered what hidden secrets it holds? Well, today we're diving deep into the fascinating world of determinants. Specifically, we've got this gnarly matrix:

$$A = \begin{pmatrix} 3x+2 & -1 & 1 \ x & 1 & 1 \ 1 & 1 & 2 \end{pmatrix}$$

And we're told its determinant is a cool 9. Our mission, should we choose to accept it, is to find the value of x! This isn't just some abstract math problem; understanding determinants is super crucial in various fields, from solving systems of linear equations to understanding transformations in geometry and even in quantum mechanics. So, let's roll up our sleeves and crack this code!

Understanding Determinants: The Key to Unlocking Matrix Secrets

Before we jump into solving for x, let's quickly recap what a determinant actually is, especially for a 3x3 matrix like the one we're dealing with. Think of the determinant as a special number that can be calculated from a square matrix. It tells us a lot about the matrix itself. For instance, if the determinant is zero, it means the matrix is singular, and it doesn't have an inverse. It also tells us about the scaling factor of the linear transformation represented by the matrix. When we're given that the determinant of our matrix A is 9, we're essentially being given a specific property that will help us isolate and solve for the unknown variable x.

Calculating the determinant of a 3x3 matrix involves a specific formula. For a general 3x3 matrix:

$$M = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix}$$

The determinant, denoted as |M| or det(M), is calculated as:

$$\det(M) = a(ei - fh) - b(di - fg) + c(dh - eg)$$

This formula might look a bit intimidating at first, but it's essentially a systematic way of combining the elements of the matrix. We pick an element from the first row, multiply it by the determinant of the 2x2 submatrix formed by removing the row and column of that element, and then alternate signs (+, -, +). It's like a little dance of multiplication and subtraction.

For our specific matrix $A = \begin{pmatrix} 3x+2 & -1 & 1 \ x & 1 & 1 \ 1 & 1 & 2 \end{pmatrix}$, we'll apply this formula. Let's identify our 'a', 'b', 'c', etc.:

• a = $3x+2$
• b = $-1$
• c = $1$
• d = $x$
• e = $1$
• f = $1$
• g = $1$
• h = $1$
• i = $2$

Now, let's plug these values into the determinant formula. It's crucial to be super careful with the signs and the algebra here, as a small slip can lead to a completely wrong answer. Remember, we're aiming to get an equation where we can solve for x, and the determinant value of 9 is our target. So, let's get ready for some algebraic manipulation!

Calculating the Determinant of Matrix A Step-by-Step

Alright, guys, let's get down to business and calculate the determinant of our matrix $A$. We're going to use the formula we just discussed: $ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) $. Remember, we're given that $\det(A) = 9$. Let's substitute the elements of matrix $A$ into the formula:

$$\det(A) = (3x+2)((1)(2) - (1)(1)) - (-1)((x)(2) - (1)(1)) + (1)((x)(1) - (1)(1))$$

Woah, that looks like a mouthful! But we can break it down. Let's tackle each part piece by piece.

First term: $(3x+2)((1)(2) - (1)(1))$

• The inner part is $(1)(2) - (1)(1) = 2 - 1 = 1$.
• So, the first term becomes $(3x+2)(1) = 3x+2$.

Second term: $- (-1)((x)(2) - (1)(1))$

• First, let's handle the double negative: $- (-1)$ becomes $+1$.
• The inner part is $(x)(2) - (1)(1) = 2x - 1$.
• So, the second term becomes $+1(2x - 1) = 2x - 1$.

Third term: $+ (1)((x)(1) - (1)(1))$

• The inner part is $(x)(1) - (1)(1) = x - 1$.
• So, the third term becomes $1(x - 1) = x - 1$.

Now, we combine all these simplified terms to get the determinant of matrix $A$:

$$\det(A) = (3x+2) + (2x-1) + (x-1)$$

Let's simplify this expression by grouping like terms (the x's and the constants):

$$\det(A) = (3x + 2x + x) + (2 - 1 - 1)$$

$$\det(A) = 6x + 0$$

$$\det(A) = 6x$$

See? We systematically broke it down, and it wasn't as scary as it looked! We've successfully calculated the determinant of matrix $A$ in terms of x, and it simplifies to a neat $6x$. This is a huge step towards finding our unknown value!

Solving for x: The Grand Finale!

We've done the heavy lifting, guys! We calculated the determinant of matrix $A$ and found that $\det(A) = 6x$. Now, we bring in the information given in the problem: the determinant of matrix $A$ is equal to 9. This means we can set up a simple equation:

$$6x = 9$$

To solve for x, we just need to isolate it. We can do this by dividing both sides of the equation by 6:

$$x = \frac{9}{6}$$

And now, we simplify the fraction. Both 9 and 6 are divisible by 3:

$$x = \frac{9 \div 3}{6 \div 3} = \frac{3}{2}$$

So, there you have it! The value of x that makes the determinant of matrix $A$ equal to 9 is $x = \frac{3}{2}$. Isn't that awesome? We took a matrix with an unknown variable, used the definition of a determinant, performed some careful algebra, and arrived at a concrete solution. This process highlights the power of mathematical formulas and systematic problem-solving.

Verification: Let's Double-Check Our Work!

It's always a good idea to verify our answer, especially in math, right? Let's substitute $x = \frac{3}{2}$ back into the original matrix $A$ and calculate its determinant to see if we get 9.

First, let's find the value of $3x+2$ when $x = \frac{3}{2}$:

$$3x+2 = 3\left(\frac{3}{2}\right) + 2 = \frac{9}{2} + 2 = \frac{9}{2} + \frac{4}{2} = \frac{13}{2}$$

Now our matrix $A$ looks like this:

$$A = \begin{pmatrix} \frac{13}{2} & -1 & 1 \ \frac{3}{2} & 1 & 1 \ 1 & 1 & 2 \end{pmatrix}$$

Let's calculate its determinant using the formula:

$$\det(A) = \frac{13}{2}((1)(2) - (1)(1)) - (-1)((\frac{3}{2})(2) - (1)(1)) + (1)((\frac{3}{2})(1) - (1)(1))$$

$$\det(A) = \frac{13}{2}(2 - 1) + 1(3 - 1) + 1(\frac{3}{2} - 1)$$

$$\det(A) = \frac{13}{2}(1) + 1(2) + 1(\frac{1}{2})$$

$$\det(A) = \frac{13}{2} + 2 + \frac{1}{2}$$

$$\det(A) = \frac{13}{2} + \frac{4}{2} + \frac{1}{2}$$

$$\det(A) = \frac{13 + 4 + 1}{2} = \frac{18}{2}$$

$$\det(A) = 9$$

Boom! It matches! Our calculated determinant is indeed 9. This confirms that our value of $x = \frac{3}{2}$ is absolutely correct. It's always super satisfying when your answer checks out, right?

Why Determinants Matter: Beyond the Classroom

So, why should you even care about determinants? As we touched upon earlier, they're not just abstract mathematical concepts confined to textbooks. Determinants have real-world applications that make them incredibly valuable tools. In computer graphics, for instance, determinants are used to determine the volume scaling factor of a transformation. If you're working with 3D models and applying rotations or scaling, the determinant of the transformation matrix tells you how the volume of an object changes.

In engineering, particularly in structural analysis, determinants are used to check if a system of equations describing a structure is solvable. If the determinant of the coefficient matrix is non-zero, the system has a unique solution, which is crucial for predicting how a structure will behave under stress.

Furthermore, in economics, determinants help in solving complex systems of linear equations that model market behavior, resource allocation, or economic forecasting. The ability to find unknown variables like x in our problem is fundamental to solving these larger, more complex models.

Even in data science and machine learning, understanding linear algebra, including determinants, is essential for algorithms that deal with high-dimensional data. Concepts like principal component analysis (PCA) rely heavily on the properties of matrices and their determinants to reduce data dimensionality while retaining important information.

So, the next time you encounter a problem involving matrices and determinants, remember that you're not just manipulating numbers; you're engaging with a powerful mathematical tool that underpins many advanced technologies and scientific fields. Keep practicing, keep exploring, and who knows what mathematical adventures await you!